The higher-rank Godbersen conjecture for Cartesian powers

Let $K\subset\mathbb{R}^n$ be a convex body. Let $\Delta_p:\mathbb{R}^n\to(\mathbb{R}^n)^p$ be the diagonal embedding, and let $\iota_i:\mathbb{R}^n\to(\mathbb{R}^n)^p$ be inclusion into the $i$-th factor. For a body $A$, write $A[r]$ for $r$ copies of $A$, and let $V$ denote mixed volume in $(\mathbb{R}^n)^p$. Higher-rank Godbersen conjecture. For any $p\in\mathbb{N}$, $0\leq k\leq n$, and $0\leq k_i\leq n$ for $i=1,\dots,p$, such that $k_1+\cdots+k_p=k$, one has

$$V(-\Delta_pK[k],\iota_1K[n-k_1],\dots,\iota_pK[n-k_p])\leq {n\choose k}{k\choose k_1,\dots,k_p}\frac{(n!)^p}{(pn)!}\operatorname{vol}_n(K)^p.$$

Moreover, if $k_i<n$ for all $i$, $0<k$, and $K$ has nonempty interior, equality holds if and only if $K$ is a simplex. This conjecture strengthens the higher-order Godbersen conjecture; the paper proves both for anti-blocking convex bodies, but the general assertion remains open.